21.15. Numerical Solutions using Linear Algebra

Try these first

1. 1.

   What is the Gauss-Seidel method?

   Hint

   Think about how variables are updated in iterations.

2. 2.

   Define a sparse matrix.

   Hint

   What is the characteristic of a matrix when it has many zeros?

3. 3.

   Which of the following describes an iterative method?

   • Calculating solutions exactly
   • Refining estimates successively
   • Using only direct calculations

   Hint

   Consider how iterative methods work.

4. 4.

   True or False: The Gauss-Seidel method updates all variables simultaneously.

   • True
   • False

   Hint

   Think about how variables are processed in this method.

5. 5.

   Given a system of equations represented in matrix form, solve it using both the Gauss-Seidel and Jacobi methods. Compare convergence speeds.

   Hint

   Set up the system and carefully apply both methods step by step.

6. 6.

   Explain how the choice of relaxation factor in SOR affects convergence. Create two scenarios with different factors and analyze the results.

   Hint

   Experiment with values less than and greater than 1 and observe the outcomes.